Computably Based Locally Compact Spaces

Paul Taylor

ASD (Abstract Stone Duality) is a re-axiomatisation of
general topology in which the topology on a space is treated,
not as an infinitary lattice,
but as an exponential object of the same category
as the original space, with an associated lambda-calculus.
In this paper, this is shown to be equivalent to a notion
of computable basis for locally compact sober spaces or locales,
involving a family of open subspaces and accompanying family
of compact ones.
This generalises Smyth’s effectively given domains
and Jung’s Strong proximity lattices.
Part of the data for a basis is the inclusion relation of compact
subspaces within open ones, which is formulated in locale theory as
the way-below relation on a continuous lattice.
The finitary properties of this relation are characterised here,
including the Wilker condition for the cover of a compact
space by two open ones.
The real line is used as a running example,
being closely related to Scott’s domain of intervals.
ASD does not use the category of sets,
but the full subcategory of overt discrete objects plays this role;
it is an arithmetic universe (pretopos with lists).
In particular, we use this subcategory to translate
computable bases for classical spaces into objects in the ASD calculus.

Acknowledgements

This paper evolved from [],
which included roughly Sections
??–?? and ?? of the present version.
It was presented at
Category Theory and Computer Science9,
in Ottawa on 17 August 2002,
and at Domains Workshop6 in Birmingham a month later.
The characterisation of locally compact objects using effective bases
(Section ??) had been announced on categories
on 30 January 2002.

The earlier version also showed that any object has a filter basis,
and went on to prove Baire’s category theorem, that the intersection
of any sequence of dense open subobjects (of any locally compact overt object)
is dense.
These arguments were adapted from the corresponding ones in the theory
of continuous lattices [, Sections I 3.3 and 3.43].

I would like to thank
Andrej Bauer,
Martn Escardó,
Peter Johnstone,
Achim Jung,
Jimmie Lawson,
Graham White
and the CTCS and LMCS referees for their comments.
Graham White has given continuing encouragement
throughout the abstract Stone duality project,
besides being an inexhaustible source of mathematical ideas.